Linear maps strongly preserving Moore-Penrose invertibility

Burgos, María; Márquez-García, A. C.; Campoy, Antonio Morales

doi:10.7153/oam-06-53

Cited by 7 publications

(4 citation statements)

References 18 publications

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“…Every Jordan * -homomorphism strongly preserves Moore-Penrose invertibility, and the question is whether or not the converse holds. Some partial positive answers are given by Mbekhta in [3], and more recently by the authors of the present paper in [12], when has a rich structure of projections. The problem for general * -algebras remains open.…”

Section: Preliminariesmentioning

confidence: 65%

Approximate Preservers on Banach Algebras andC*-Algebras

Burgos¹,

Márquez-García

Campoy

2013

Abstract and Applied Analysis

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Section: Preliminariesmentioning

confidence: 65%

Approximate Preservers on Banach Algebras andC*-Algebras

Burgos¹,

Márquez-García

Campoy

2013

Abstract and Applied Analysis

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“…In the next proposition we show that every Jordan * -homomorphism preserves the diamond partial order in the setting of all regular elements. It can be proved by using [6,Remark 8] and [9,Proposition 3.1]. However we include its proof here for the sake of completeness.…”

Section: Maps Preserving the Diamond Partial Ordermentioning

confidence: 99%

Maps preserving the diamond partial order

Burgos¹,

Mrquez-Garca

Campoy

2017

Applied Mathematics and Computation

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“…In [33], M. Mbekhta proved that a surjective unital bounded linear map from a real rank zero C * -algebra to a prime C * -algebra strongly preserves Moore-Penrose invertibility if and only if it is either an * -homomorphism or an * -anti-homomorphism. Recently in [8] the first three authors of this note show that a linear map T strongly preserving Moore-Penrose invertibility between C * -algebras A and B, is a Jordan * -homomorphism multiplied by a regular element of B commuting with the image of T , whenever the domain C * -algebra A is unital and linearly spanned by its projections, or when A is unital and has real rank zero and T is bounded. It is also proved that every bijective linear map strongly preserving Moore-Penrose invertibility from a unital C * -algebra with essential socle is a Jordan * -isomorphism multiplied by an involutory element.…”

Section: Introductionmentioning

confidence: 99%

Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers

Burgos¹,

Márquez-García²,

Campoy³

et al. 2016

Banach J. Math. Anal.

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We study new classes of linear preservers between C * -algebras and JB * -triples. Let E and F be JB * -triples with ∂ e (E 1 ). We prove that every linear map T : E → F strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C * -algebras A and B, for each linear map T strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan * -homomorphism S : A → B satisfying T (x) = T (1)S(x), for every x ∈ A. We also study the connections between linear maps strongly preserving Brown-Pedersen quasi-invertibility and other clases of linear preservers between C * -algebras like Bergmann-zero pairs preservers, Brown-Pedersen quasiinvertibility preservers and extreme points preservers.2010 Mathematics Subject Classification. 47B49, 15A09, 46L05, 47B48.between Banach algebras is a Jordan homomorphism if T (a 2 ) = T (a) 2 , for all a ∈ A (equivalently, T (ab + ba) = T (a)T (b) + T (b)T (a), for all a, b ∈ A). If A and B are unital, T is called unital if T (1) = 1. If A and B are C * -algebras and T (a * ) = T (a) * , for every a ∈ A, then T is called symmetric. Symmetric Jordan homomorphisms are named Jordan * -homomorphisms.Consequently, the problem of studying the linear maps T : A → B such that T (∂ e (A 1 )) ⊆ ∂ e (B 1 ) is a more general challenge, which is directly motivated by the just mentioned consequence of the Russo-Dye theorem. We only know partial answers to this problem. Concretely, V. Mascioni and L. Molnár studied the linear maps on a von Neumann factor M which preserve the extreme points of the unit ball of M. They prove that if M is infinite, then every linear mapping T on M preserving extreme points admits a factorization of the form T (a) = uS(a) (a ∈ M), where u is a (fixed) unitary in M and S either is a unital * -homomorphism or a unital * -anti-homomorphism (see [30, Theorem 1]). Theorem 2 in [30] states that, for a finite von Neumann algebra M, a linear map T : M → M preserves extreme points of the unit ball of M if and only if there exist a unitary operator u ∈ M and a unital Jordan * -homomorphism S : M → M such that T (a) = uS(a) (a ∈ A). In [29], L.E. Labuschagne and V.Mascioni study linear maps between C * -algebras whose adjoints preserve extreme points of the dual ball.The above results of Mascioni and L. Molnár are the most conclusive answers we know about linear maps between unital C * -algebras preserving extreme points. In this note we shall revisit the problem in full generality. We present several counter-examples to illustrate that the conclusions proved by Mascioni and Molnár for von Neumann factors need not be true for linear mappings preserving extreme points between unital C * -algebras (compare Remarks 5.8 and 5.9). It seems natural to ask whether a different class of linear preservers satisfies the same conclusions found by Mascioni and Molnár.Every unital Jordan homomorphism between Banach algebras strongly preserves invertibility, that is, T (a −1 ) = T (a) −1 , ...

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Linear maps strongly preserving Moore-Penrose invertibility

Cited by 7 publications

References 18 publications

Approximate Preservers on Banach Algebras andC*-Algebras

Approximate Preservers on Banach Algebras andC*-Algebras

Maps preserving the diamond partial order

Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers

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